A164908 a(n) = (3*4^n - 0^n)/2.

1, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736, 6442450944, 25769803776, 103079215104, 412316860416, 1649267441664, 6597069766656, 26388279066624, 105553116266496, 422212465065984, 1688849860263936

Offset: 0

Author

Klaus Brockhaus, Aug 31 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A164907. Inverse binomial transform of A057651.

Partial sums are in A083420.

Decimal representations of the n-th iterations of elementary cellular automata rules 14, 46, 142 and 174 generate this sequence (see A266298 and A266299). - Karl V. Keller, Jr., Aug 31 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
• Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
• N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2.
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
• Index entries for sequences related to cellular automata.
• Index entries for linear recurrences with constant coefficients, signature (4).

Crossrefs

Equals 1 followed by A002023 (6*4^n). Essentially the same as A084509.

Cf. A164907, A057651, A083420 (2*4^n-1), A247640, A266298, A266299.

Programs

Magma

[ (3*4^n-0^n)/2: n in [0..22] ];

Mathematica

a[n_]:=(MatrixPower[{{2,2},{2,2}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Join[{1},(3*4^Range[25])/2] (* or *) Join[{1},NestList[4#&,6,25]] (* Harvey P. Dale, Feb 14 2012 *)

PARI

a(n)=3*4^n\2 \\ Charles R Greathouse IV, Oct 12 2015

Python

print([int(6*4**(n-1)) for n in range(50)]) # Karl V. Keller, Jr., Aug 30 2021

Formula

a(n) = 4*a(n-1) for n > 1; a(0) = 1, a(1) = 6.

G.f.: (1+2*x)/(1-4*x).

a(n) = floor(6*4^(n-1)). - Karl V. Keller, Jr., Aug 30 2021

E.g.f.: (3*exp(4*x) - 1)/2. - Elmo R. Oliveira, Mar 31 2025